Notations

 

number of crop types;

number of planting dates for crop k;

= number of insurance policies for crop k;

number of scenarios;

planting area (in acres) available for crop k;

planting cost per acre of crop k;

premium of insurance policy i for crop k per acre;

number of acres of land for crop k planting on date t and insured by policy i;

additional  variable indicating selection of insurance policy i for crop k;

(random) revenue of crop k per acre planted on date t and insured by policy i;

revenue of crop k per acre planted on date t and insured by policy i under scenario j;

threshold of losses in probability exceeding penalty constraint;

the upper bound of the probability exceeding penalty constraint;

upper bound on

upper bound on

yield of crop k per acre planted on date t under scenario j;

(historical) average yield of crop k per acre;

market price of crop k per pound under scenario j;

price base of crop k per pound.

 

Optimization Problem

 

Maximizing expected profit

 

subject to

 

planting area constraint for each crop k

 

joint constraint on planting area and insurance policy

 

each crop k can be insured by at most one policy

 

 

risk constraint (CVaR, VaR, and Probability Exceeding Penalty)

 

or

or

 

constraint on additional  variables

 

lower bounds on variables